SHORT COMMUNICATION
Parametric study of acoustic focusing of particlesin a micro-channel in the perspective to improvemicro-PIV measurements
Olivier Dron Æ Claire Ratier Æ Mauricio Hoyos ÆJean-Luc Aider
Received: 27 March 2009 / Accepted: 27 June 2009 / Published online: 30 July 2009
� Springer-Verlag 2009
Abstract In this study, we propose a new technique to
bring most of the particles in a horizontal plane at mid-
height of a micro-channel in order to improve the quality of
the micro-PIV measurements. The basic principle is to
create a stationary acoustic wave along the channel height
so that the resulting acoustic force moves the particles
toward the pressure node. A parametric study has been
carried out without mean flow to characterize the motion
of the particles toward the nodal-plane. We found that
focusing speed grows with the acoustic pressure amplitude,
with the concentration of particles in the suspension and
with the particles diameter. We also led a preliminary
investigation of acoustic focusing together with a mean
stationary flow. We still observed an important focusing of
particles for low freestream velocities. Nevertheless,
acoustic focusing is inefficient beyond a given critical
freestream velocity U0 for a given set of acoustic param-
eters and a given type of particles. It was also shown that
other phenomena, like clumps formation, can be observed
without mean flow if the acoustic focusing lasts long
enough.
Keywords Flow field velocimetry � Acoustic streaming �Standing wave
1 Introduction
While the last 10 years have seen enormous progress in
the general areas of Micro Electro Mechanical Systems
(MEMS) design and fabrication (Gad-el-hak 1999), the
understanding of fundamental physical processes on these
small scales is hampered by the lack of suitable quantita-
tive measurement and simulation tools.
Velocity measurements at small scale is a challenge and
requires dedicated techniques. Several common macro-
scopic measurement techniques have been extended to
microscopic length scales, like micro-LDV (Laser Doppler
Velocimetry) (Thieu et al. 1995), micro-MTV (Molecular
Tagging Velocimetry) (Garbe et al. 2008), and micro-
PIV (Particle Image Velocimetry) (Santiago et al. 1998;
Meinhart and Santiago 1999). Micro-LDV will give accu-
rate time dependant velocity measurements in a given point
of the volume, while the micro-MTV technique allows
mainly for velocity measurements along a line.
For in-plane velocity measurements, one has to use
micro-PIV, which is actually similar to the macroscopic
PIV method, which means bulk seeding and auto-correla-
tion calculations to get velocity fields. However, the laser
sheet illumination is irrelevant for micro-PIV since the laser
sheet thickness (the laser Waist) and the micro-channel
height are roughly of the same value. Instead, the whole
volume of the flow is illuminated at the same time, which
means all the tracer particles are illuminated, even the ones
that are not in the measurement plane. Those particles
generate background noise on the images, and lower the
resolution. Several techniques to improve micro-PIV have
O. Dron (&) � J.-L. Aider
Team ‘‘Instability, Control and Turbulence’’ ESPCI,
Laboratoire de Physique et Mecanique des Milieux Heterogenes,
10 rue Vauquelin, Paris Cedex 05 75231, France
e-mail: [email protected]
J.-L. Aider
e-mail: [email protected]
C. Ratier � M. Hoyos
Team ‘‘Separation’’ ESPCI, Laboratoire de Physique et
Mecanique des Milieux Heterogenes, 10 rue Vauquelin,
Paris Cedex 05 75231, France
123
Microfluid Nanofluid (2009) 7:857–867
DOI 10.1007/s10404-009-0477-7
already been established, based on image processing after
acquisition (Bourdon et al. 2004) or hydrodynamic focus-
ing (Mielnik and Saetran 2006). In this article, we describe
an alternative method to create thin sheet of particles in the
micro-channel instead of using a laser sheet to illuminate
particles evenly distributed throughout the whole channel.
The method consists in focusing the particles using acoustic
force created by an acoustic standing wave along the
height of the channel. Before using acousting focusing to
improve micro-PIV measurements, it is necessary to study
the dynamics of focalization without mean-flow and its
dependance on different physical parameters. This is the
objective of the present article which is organized as
follows.
First, the experimental setup and experimental tech-
niques are presented. Then, the principle of acoustic
focusing of particles is explained and some characteristic
times are derived from the equation governing the particles
motion. A detailed parametric study of acoustic focusing
without mean flow is then described before showing an
example of acoustic focusing together with a mean sta-
tionary flow. Finally, some other phenomena observed after
long focusing times are shortly described in the discussion
before turning to the conclusion.
2 Experimental setup and methods
2.1 Flow facility
The experiments were carried out using a 0.36 mm (height
h) 9 3 mm (width) 9 40 mm (length) micro-channel. The
channel is composed of one mylar sheet stuck between two
1 mm plexiglas plates. The drawing of the micro-channel is
cut into the mylar sheet so that the channel height is
imposed by the mylar sheet thickness (0.36 mm in this
case). The micro-channel is designed for low velocity
flows, but the velocities are large enough to study acoustic
focusing and its limits. The height of the channel is chosen
to fit an acoustic purpose that will be described later. The
fluid used for these experiments was distilled water.
2.2 Micro-PIV setup
As shown on Fig. 1, a 15 Hz pulsed Nd:YAG laser illu-
minates the fluorescent flow-tracing particles through an
epi-fluorescent inverted microscope. Fluorescent particles
absorb the green light (k & 532 nm) and reemit the light
to a slightly higher wavelength (k & 560 nm). To reduce
the noise induced by the incoming wavelength, a filter cube
is used with a narrow filter band around k & 560 nm so
that only the fluorescent light is recorded by the double-
frame CCD camera. The micro-PIV set-up is the same as
the one proposed by Meinhart and Santiago (1999).
The image resolution is 1200 9 1600 pixels with a 12
bits dynamic range and a sampling frequency of f = 15 Hz.
The whole PIV process from acquisition to vector post-
processing is carried out with Davis 7.1 software from
LaVision�.
2.3 Seeding particles
The tracer particles for the l-PIV are red fluorescing
polymer microspheres 1% solids, with a 2–7 lm diameter
ðUÞ: It is small enough for the particles (with a specific
density of qp = 1.05 g cm-3) to follow the flow faithfully
without perturbing it, and large enough for them to scatter
sufficient light without being affected by Brownian motion.
We indeed observed that the Brownian motion of those
particles does not affect velocity measurements. These
particles are doped with a fluorescent dye with a peak
excitation wavelength of kabs & 532 nm and a peak
emission wavelength of kemit & 560 nm. The choice of the
initial concentration of flow-tracing particles comes from
auto-correlation techniques criteria and is also one of the
parameters that will be varied in the following.
The flow is produced by a syringe pump which allows
flow rates from 1–99 times a reference flow rate Q01
= 0.7 ml h-1. It corresponds to velocities ranging from
U0 = 0.18 mm s-1 to 18 mm s-1, leading to a Reynolds
number based on the micro-channel height ranging from
Reh ¼ hUm � 0:06 to 6, where m = 10-6 m2 s-1 is the
kinematic viscosity of water. As expected the flow is in the
laminar regime.
Fig. 1 Description of the micro-PIV setup. Flow tracing particles are
illuminated by two pulsed Nd:YAG lasers through an epi-fluorescent
inverted microscope. The particle images are recorded by a double-
frame CCD camera (LaVision� Adapted from Meinhart and Santiago
(1999)
858 Microfluid Nanofluid (2009) 7:857–867
123
2.4 Acoustic setup
An 8 mm diameter transducer, whose functional frequency
is roughly fac = 2 MHz, is used to generate the acoustic
wave. The voltage applied to the transducer is the param-
eter used to control the acoustic amplitude. As shown on
Fig. 2a, the transducer is set on one side of the micro-
channel, while the inverted microscope is on the other side.
Thanks to this setup, one can see the motion of the particles
right under the acoustic transducer which is necessary to
characterize the dynamics of the focusing of the particles
with or without mean flow.
3 Acoustic focusing
3.1 Theory
3.1.1 Principle
The main objective of this study is to demonstrate and
characterize the acoustic focusing of particles in a micro-
channel which could be a very useful phenomenon to
improve micro-PIV visualizations. Of course, acoustic
focusing can also be helpful in any situations where one
wants to move particles in a given direction without
changing the mean flow. The acoustic radiation force,
which is the origin of the acoustic focusing, has already
been used to manipulate particles, such as the formation of
particle lines (Nield et al. 2007; Oberti et al. 2009) or the
separation of particles as a function of their physical
properties (Hoyos et al. 2005; Petersson et al. 2005).
The method is based on the creation of a standing
acoustic wave along the height of the micro-channel to
move the tracer particles toward the acoustic pressure
nodal-plane (Fig. 2b). This will be possible thanks to the
non-linearity of the acoustic force. Indeed, considering
only the linear term of the acoustic force, leads to a time-
average of the acoustic force on the particles equal to zero,
unlike non-linear terms. In order to identify and quantify
the relevant parameters of the phenomenon, one has to
study the acoustic focusing dynamics without mean flow.
In the following, we describe the dynamics of a single
particle in still water and in an acoustic field.
Theoretically, one has to consider a viscous fluid. It
is appropriate to neglect viscous effects as long as the
particle diameter is smaller than the acoustic wavelength
U� kð Þ but larger than the acoustic boundary layer
U� d; with d ¼ mpfac
� �0:5� �
(Doinikov 1996). In our
experiments, d = 0.4 lm, k = 720 lm, and 2 lm�U�7 lm so that the viscous effects are negligible.
The acoustic wave will also have another consequence on
the particles: it will create acoustic streamings that will drag
the particles thanks to the Stokes force (see Sect. 6.2). It is
possible (Sundin et al. 2007), setting the system parameters,
especially the particle size, to make one of the two
effects dominant over the other one. In our experiments, the
acoustic force is dominant over acoustic streaming.
3.1.2 Forces
Particles in motion in still water and in an acoustic field are
subjected to three forces (Fig. 2b):
– The acoustic force FA as defined by Gor’kov (1962),
– The buoyancy FB, and
– The Stokes force FS.
The analytic expressions of those forces are:
FA�! ¼ Vpk EA x; yð Þh i A1 þ
3
2A2
� �sin 2kzð Þez
! ð1Þ
FB�! ¼ �Vp qp � qf
� �gez! ð2Þ
Fig. 2 a A syringe pump
injects distilled water seeded
with particles into the 360 lm
micro-channel. The particles are
exposed to a standing wave
induced by the acoustic
transducer. In the same time,
they are also illuminated and
observed by the micro-PIV
system. b Description of the
forces applied on a particle in
still water and in standing
acoustic wave. The acoustic
force pulls the particle toward
the acoustic pressure node
against the Stokes force and
buoyancy
Microfluid Nanofluid (2009) 7:857–867 859
123
FS�! ¼ 6plR U
!x; zð Þ � u! x; zð Þ
� �ð3Þ
where R is the particle radius, Vp ¼ 43pR3 is the particle
volume, k is the acoustic wave number (andk the wavelength),
EA is the acoustic wave energy (h i stands for the time-
average), qp is the particle density and qf the fluid density, l is
the dynamic viscosity, and U!
and u! are, respectively, the
flow velocity and the particle velocity in the channel frame.
The two coefficients A1 and A2 depends on the density and
sound velocity in both the fluid and the particles:
A1 ¼ 1� c2f qf
c2pqp
; A2 ¼2 qp � qf
� �
2qp þ qf
ð4Þ
In the following experiments, the fluid is water (qf =
1000 kg m-3) and the particles are made of polymer
(qp = 1050 kg m-3). The sound velocity in the fluid and in
the particles are cf = 1435 m s-1 and cp = 2407 m s-1,
respectively. Considering the given physical parameters,
one gets A1 þ 32A2
� �� 0:71 [ 0:
If one considers z = 0 at the bottom of the channel, and
positive upward, then the acoustic force is directed toward
the nodal-plane. When A1 þ 32A2
� �is negative, then the
acoustic force is directed toward the pressure anti-nodal-
plane (Spengler et al. 2003). In the following q will be
used instead of qp, and Dq ¼ qp � qf :
Here, one should emphasize that the acoustic force
defined in Eq. 1 has been derived for an ideal fluid
(Gor’kov 1962). For actual fluids, one should take into
account thermal and viscous dissipation in the derivation
of the acoustic force as proposed by Doinikov (1996). If
we compute with our physical parameters the acoustic
(ka = 720 lm), thermal (kt = 0.15 lm), and viscous
(kv = 0.4 lm) wavelengths as defined by Doinikov, we see
that ðka � kt; kv;/Þ showing that our experimental con-
ditions correspond to the long sound wavelength limit and
weak dissipative effects (kv, kt \ /).
3.1.3 Sedimentation and focusing times
In order to determine characteristic sedimentation and
focusing times, one can integrate the law of resultant force
projected along the z-axis, without the acoustic force for
the sedimentation time, and with the acoustic force for the
focusing time. It leads to the following equation:
1
3U2q€z ¼ �6l _z� 1
3U2Dq g
þ 1
3U2k EA x; yð Þh i A1 þ
3
2A2
� �sin 2kzð Þ ð5Þ
where U is the particle diameter.
Initial conditions are _zðt ¼ 0Þ ¼ 0 (no initial speed)
for the velocity, kzðt ¼ 0Þ ¼ p2
for the sedimentation (the
particle is located at the mid-height of the channel)
and kzðt ¼ 0Þ ¼ p for the focusing (the particle is
located at the top of the channel). One can define the
sedimentation (TS) and focusing (TF) times such as the
time needed for a particle to travel half the height
channel, h:
kzðt ¼ TSÞ ¼h
2and kzðt ¼ TFÞ ¼
h
2:
It leads to the following expressions and numerical values
for our experiments:
TS ¼9lh
U2Dqgð6Þ
TF ¼9
2
lk
U2k EA x; yð Þh i A1 þ 32A2
� � ð7Þ
Using the physical parameters of the present experiment,
we obtain TS & 1340 s and TF & 2.6 s, with a 10% error
for U ¼ 2:2 lm ð�5%Þ; TS � 674 s and TF & 1.3 s, with a
10% error for U ¼ 3:1 lm (±5%) and TS & 132 s and
TF & 0.25 s, with a 36% error for U ¼ 7 lm (±18%). The
way EAðx; yÞh i is estimated experimentally is detailed in
Sect. 6.2. One can conclude from those numerical values
that sedimentation is negligible compared to acoustic
focusing for short times.
The mean focusing times correspond to a mean focusing
velocity uF ranging between 60 and 792 lm s-1. The
Reynolds numbers associated to the motion of the particles
induced by acoustic force ReU ¼ UUm ranges between 10-5
and 4.6 9 10-4 for the three diameters so that there should
not be any wake or added mass effect.
3.2 Acoustic frequency
In the following, the acoustic frequency is tuned to induce
a standing wave with only one pressure nodal-plane along
the height of the micro-channel so that only one focusing
plane will be considered. The micro-channel height cor-
responds to half of the wavelength in that case.
4 Acoustic focusing without mean flow
The main goal of this study is to confirm and to quantify
the acoustic focusing of particles in a micro-channel
without mean flow as a first step before application to
micro-PIV measurements. The second step consists in
studying the acoustic focusing of particles in a stationary
mean flow. It is illustrated by very preliminary results in
the second part of the article. The influence of the fol-
lowing parameters has been studied without mean flow:
860 Microfluid Nanofluid (2009) 7:857–867
123
the acoustic amplitude (through the voltage applied on the
acoustic transducer), the particles concentration, and the
diameter of the particles.
4.1 Observation of acoustic focusing
The focusing dynamics is illustrated in Fig. 3 through
different snapshots of the particles ðU ¼ 2 lmÞ in the
nodal-plane, which is merely the camera focal plane, for
different successive time steps. At time t = 0, the trans-
ducer is turned on. Particles that are located in the focal
plane are seen as white dots. Out-of-plane particles appear
as diffuse spots. The more distant from the focal plane the
particles are, the larger the spots. One can clearly see on
Fig. 3b–d that the number of particles in the nodal-plane is
increasing as a function of time. Actually, one could even
see the diffuse spots getting smaller while the particles
move toward the nodal-plane. After some time, the
focusing slows down, mainly because most of the particles
are already in the nodal-plane (Fig. 3d).
We can conclude from Fig. 3 that the order of magnitude of
the experimental focusing time, with those physical parame-
ters, is around a hundred of seconds. This is a lot more than the
numerical value TF & 2.6 s found in Sect. 3.1, whereas the
parameter values are almost the same. First of all, one has to
keep in mind that the two values merely mean something
different. TF is the time needed for a single particle to move
from the top of the channel to its middle plane. The interac-
tions between particles is neglected, as well as any viscous or
screening effect which could lower the acoustic focusing. If
one can expect some dissipation of acoustic energy, the
acoustic wave might also not be purely a standing wave, but
could be for a small part a traveling wave and a larger part a
standing wave. All those phenomena are difficult to quantify
but all could contribute to slow down the dynamics of acoustic
focusing.
Fig. 3 Time-evolution of the particles ðU ¼ 2 lmÞ concentration in
the nodal-plane. a The focusing has not started yet. The particles
distribution along z is homogeneous. We can see more out-of-plane
particles than in-plane particles. At time t = 0, a 2 MHz acoustic wave
(P = 5 V) is generated. b t = 30 s, c t = 65 s, and d t = 199 s—the
number of focused particles increases with increasing time
Microfluid Nanofluid (2009) 7:857–867 861
123
4.2 Parametric study of the acoustic focusing
A way to quantify the dynamics of the focusing is to
evaluate the number of particles in the nodal-plane as a
function of time. The method used to calculate the number
of particles is the following. Using the software Image J�
we first smoothen the picture. Then we use a filter to
homogenize the number of pixels defining each particle.
Finally, we divide the number of pixels that correspond to
particles by the average number of pixels for one particle.
4.2.1 Influence of the concentration of particles
The particles concentration is an important parameter for
the micro-PIV measurements as it is related to the number
of out-of-plane particles. Note that its influence does not
appear in the analytical expressions (Eqs. 6, 7) merely
because we considered dynamics for a single isolated
particle in the micro-channel, i.e., no collective effects
were taken into account in the force balance. An additional
term would be necessary to take into account the effect of
the concentration. In the following, we set C = 1 as the
concentration resulting of diluting one drop of the particles
with 10 ml of distilled water, rather than giving the exact
value of the concentration. C = 1 is then equivalent to
C = 5 9 106 particles in 1 ml of the resulting solution.
Since we are dealing with number of particles, we have
to normalize the data to make comparison easier. In the
following, the percentage of focused particles will be
considered rather than the number of focused particles.
To study the influence of the concentration, time-evo-
lution of the percentage of particles in the nodal-plane are
plotted for two different initial concentration values, C = 1
and C = 2, respectively (Fig. 4). A concentration too high
would make it impossible to count the particles, while a
low concentration would make the auto-correlation calcu-
lation difficult. This is the reason why we could not study a
wider range of concentration values. As expected, the ini-
tial concentration of particles play an important role in the
focusing dynamics. Except for the first seconds, the per-
centage of particles in the nodal-plane grows linearly with
time. One can clearly see in Fig. 4, and from the growth
rates obtained from the linear fits, that the focusing is faster
for C = 2: it is almost twice as fast as for C = 1. This
result is rather surprising, as one could have expected a
decrease of acoustic focusing when increasing the con-
centration. Unfortunately, it cannot be interpreted in the
framework of Sect. 3.1 where only a single, isolated par-
ticle is considered. It is clearly necessary to include col-
lective effects in the modeling. One possibility would be to
include a new term in Eq. 5, taking into account interaction
forces between particles. Another possibility could be to
describe the dynamics of the distribution of particles along
the height of the channel instead of following a single
particle. In this hypothesis, it should be possible to propose
a model for the dynamics of the concentration profile as a
function of time n(t, z), taking into account the transition
from a nearly homogenous vertical distribution profile to a
nearly gaussian profile as shown later in Sect. 5.1.
4.2.2 Influence of the acoustic pressure amplitude
The real pressure amplitude value is not known, but we do
control the voltage applied on the transducer. If the trans-
ducer and the whole acoustic system are linear, then the
relationship between the pressure amplitude and the volt-
age supply is also linear. In the following, the parameter
called P is the voltage value in Volt, and is proportional to
the acoustic pressure. Three different values of the acoustic
pressure will be considered: P = 3 V, P = 5 V, and
P = 10 V. P = 10 V is actually the maximum of the
voltage applied on the transducer, while for P \ 3 V
acoustic effects are barely observable.
EA, the acoustic energy density, is a quadratic function
of P, thus according to Eq. 7 one can expect a focusing
velocity proportional to P2. Results are shown in Fig. 5.
The influence of the acoustic pressure on the focusing
velocity seems to be even stronger than a quadratic one and
has to be maximized in order to get the most powerful
acoustic focusing. One could use some transducers that
support higher voltage values, but as one will see in Sect. 6,
some limitations appear when a high acoustic pressure is
used (Fig. 6).
4.2.3 Influence of the radius of the particles
The acoustic force (Eq. 1) is a cubic function of the particle
radius. Therefore, there is no doubt that this parameter has
a strong influence on the acoustic focusing. Unfortunately,
Fig. 4 Influence of the particle concentration. On the graphics is
represented, for different values of particle concentration, the
percentage of in-plane particles as a function of time. P = 5 V and
U ¼ 7 lm for every experiment
862 Microfluid Nanofluid (2009) 7:857–867
123
this parameter is not as easy to change as the other
parameters. Only a limited range of radius is manufactured.
In particular no particles are manufactured in the range
between 3 and 7 lm. This is the reason why there is only
three different particle sizes in this study.
The focusing is way faster with U ¼ 7 lm particles, but
there is nearly no difference between U ¼ 2 lm and U ¼3 lm: The result is quite reproducible. It appears that the
quadratic dependency on the radius (Eq. 7) is not properly
recovered in the experiment. It might be the consequence
of a transition between two different regimes, like what Li
et al. (2008) demonstrates for a droplet. Also, Bjerknes
forces (see end of Sect. 6.2), which is not a quadratic
function of the particle size, could interfere with the
acoustic radiation force. In the end, three values of the
parameter U are not enough to conclude. Further investi-
gation with other radius are needed to identify the real
dependency of the particle diameter on the focusing, and
possibly different regimes.
5 Acoustic focusing with a mean stationary flow
5.1 Acoustic focusing velocity limitation
One of the motivations in studying acoustic focusing is to
use it to improve the quality of the PIV velocity fields. The
next step is then naturally to check the efficiency of
acoustic focusing of particles in a stationary flow. To do so,
the parameters are fixed to P = 10 V and C = 1 in order to
optimize the acoustic focusing, according to the previous
results without mean flow. Figure 7 shows for different
flow rates the concentration profiles. Even if there is not yet
precise criterion to determine if the focusing is efficient,
one can guess that the focusing will lead to a significant
improvement of the velocity field measurements only for
Q = 0.7 and Q = 1.4 ml h-1 which lead to peaked particle
concentration profiles in the nodal-plane region. It leads us
to a critical freestream velocity U0 & 0.4 mm s-1 for these
experimental conditions ðP ¼ 10 V;C ¼ 1; and U ¼ 2 lmÞ:The efficiency of the method is actually the result from
the competition between the focusing time (time needed for
a particle to arrive in the nodal-plane) and the time of flight
(time needed for a particle to cross the region influenced by
the acoustic field). If the particles velocity is too large, then
the particles will not have the time to reach the nodal-
plane.
5.2 Preliminary investigation of the influence
of acoustic focusing of particles on PIV fields
In Fig. 8, two velocity fields measured in the same
experimental conditions and post-processed exactly the
same way, are compared. Figure 8b is obtained with a
given acoustic wave (P = 10 V, 2 MHz) while Fig. 8a is
obtained without any acoustic focusing. Both PIV fields are
actually an average of 345 instantaneous velocity fields.
Fig. 5 Influence of the acoustic pressure amplitude. On the graphics
is represented, for different values of the acoustic pressure amplitude,
the percentage of in-plane particles as a function of time. C = 1 and
U ¼ 7 lm for every experiment
Fig. 6 Influence of the particle diameter. On the graphics is
represented, for different values of the particle diameter, the
percentage of in-plane particles as a function of time. C = 1 and
P = 5 for every experiment
Fig. 7 Each curve represents the vertical distribution of the particles
for different values of the flow rate. The reference (without acoustic
wave: P = 0) curve has been calculated for Q ¼ 2:1 ml h�1: C ¼ 1;P ¼ 10 V and U ¼ 2 lm for every experiments
Microfluid Nanofluid (2009) 7:857–867 863
123
The mean flow is stationary and goes from the left to the
right. Since the Reynolds number of the flow is low
(Reh = 0.06 \ 1), the flow in the microchannel is laminar
and fully developed. Therefore, one can expect the velocity
field to be a uniform set of identical arrows for a given
height. Two main observations can be made from Fig. 8.
The first observation is that there are more wrong vectors
(not validated and not shown on the mean PIV field) and
vectors with a direction that is not the expected horizontal
direction from left to right, in Fig. 8a than in Fig. 8b. The
second observation is that the velocity vectors are longer in
Fig. 8b than in Fig. 8a. It means that the mean velocity in
the focal plane is larger when the particle are focused
whereas the flow rate applied is exactly the same in both
cases. These two observations are actually the conse-
quences of both increasing number of in-plane particles
and decreasing number of out-of-plane particles.
The autocorrelation calculation that gives us the velocity
fields needs a minimal concentration of particles, or else it
leads to many errors. The particles density in Fig. 8a is
clearly too low for a proper PIV autocorrelation calcula-
tion. The calculation is improved in Fig. 8b, but it may be
even better by increasing the particle concentration of the
solution.
When the particles are not focused, they still can be seen
by the camera (Airy disks) and taken into account during
the auto-correlation computation. Since the Reynolds
number is very low (Reh = 0.06 \ 1), the flow in the mi-
crochannel follows the Poiseuille law, which is a parabolic
law with its maximum at mid-height. Therefore, out-of-
plane particles are slower than in-plane particles and leads
to an underestimation of in-plane velocities. Finally, the
right picture is much closer to the expected velocity field
than the left one. This result clearly demonstrates the
potential of the acoustic focusing to improve micro-PIV
measurements.
From a practical point of view, one should carry out the
PIV measurements downstream of the region where the
acoustic wave is applied, where the flow is not perturbed by
the acoustic focusing. The sheet of particles is created
under the transducer and the PIV measurements can be
carried out downstream on the unperturbed flow (like the
added vertical component due to the acoustic focusing).
Sedimentation (TS & 1340 s see Sect. 3.1) and Brownian
motion [TB & 105 s according to the Einstein formula
(Einstein 1905)] are too large to disperse the particles.
Therefore, particles stay focused as long as they are in the
micro-channel.
6 Discussion
6.1 Limitation of the technique
In the previous section the principle of acoustic focusing
has been validated and quantified. It has also been shown
that it can improve PIV velocity fields under some
hypothesis and using the proper parameters. It also high-
lighted some ways to optimize the focusing with different
parameters. As we already mentioned for the pressure
dependence (see Sect. 4.2), some phenomena described
below (see Sect. 6.2) prevents us to increase particle radius
to improve the velocity fields, in addition of auto-correla-
tion issues. Another way to improve the velocity fields
would be to use several transducers successively along the
streamwise direction. This way, the focusing time would
remain the same while the time of flight would increase.
This way the method could be used for larger freestream
velocities.
6.2 Streaming and structures
For the typical dimensions of our micro-fluidics device,
there are two main acoustic effects that can be expected,
because of the non-linearity of the acoustic wave:
– the acoustic radiation force (Gor’kov 1962), which
moves particles toward pressure nodes,
– acoustic streaming (Spengler and Jeckel 2001), which
induces motion onto the entire solvent.
The acoustic radiation force has already been discussed
in the previous sections so that we will focus on streaming
Fig. 8 Height-centered velocity
fields for stationary
Q = 0.7 ml h-1 flows.
aVelocity field obtained without
any acoustic wave. b A 2 MHz
acoustic wave is applied, clearly
improving the velocity field
obtained without acoustic wave
864 Microfluid Nanofluid (2009) 7:857–867
123
effects in the following. There are three different types of
acoustic streaming (Zarembo 1971; Nomura et al. 2000):
Eckart (if h� k), Schlichting (if h� k), and Rayleigh
streamings (if h & k). Only the latest has been observed
during the experiments without mean flow.
Indeed, Spengler and Jekel (2000) demonstrate that a
short container dimension along the sound axis effectively
minimizes large-scale streaming such as Eckart streaming.
Since the dimension along the sound axis in our experi-
ments is h ¼ k2; then the Eckart streaming is negligible
compared to other phenomena in our system. As for the
Schlichting streaming, it mainly consists in a vortex flow in
the acoustic boundary layer thickness d. In our experi-
ments, d = 0.4 lm. The Schlichting streaming happens
only in a very thin part of the micro-channel, therefore it is
not responsible for what is observed in our experiments.
In the Sect. 3.1, it was recalled that the acoustic force
was larger than acoustic streamings considering the phys-
ical parameters defining our experimental setup. One
would want to confirm this assertion. Rayleigh streaming
has a y-component and a z-component. Only the latest can
affect acoustic focusing. Since the channel width is wider
than the acoustic wavelength (l� k) and the acoustic
wavelength is longer than the acoustic boundary layer
thickness (k� d), the z-component has the following form
(Rayleigh 1894):
uzðy; zÞ ¼ �3v0
16c0
1� 3y2
ðl=2Þ2
!sin
4pz f
c0
� �ð8Þ
Figure 9 shows the streaming pattern calculated
according to Eq. 8. The flow generated by the acoustic
standing wave through Rayleigh streaming is directed
toward the nodal-plane for y 2 � l2ffiffi3p ; l
2ffiffi3p
h i: It is directed
away from the nodal-plane for y 2 � l2;� l
2ffiffi3p
h iand y 2
l2ffiffi3p ; l
2
h i: Assuming the particle would be dragged by the
streaming, which is highly likely since their properties have
been chosen so that inertia plays no role, the migration
caused by Rayleigh streaming would be different from
the acoustic focusing studied above. In particular, the
dependance on some physical parameter is different, like a
linear dependance on the particle radius. Thanks to Eq. 9
and experimental observations (Fig. 10a), we obtain
uzj j � 3 lm s�1; which gives us a characteristic focusing
time, such as defined in the Sect. 3.1, TF * 60 s. It is
approximatively 20 times longer than the focusing time
calculated with the acoustic radiation force. The acoustic
force is therefore much stronger over than the acoustic
streaming when it comes to acoustic focusing with
the parameters used in our experiments. Nevertheless,
Rayleigh streaming may contribute for a small part to the
observed focusing.
In the long time limit, most of the particles are
focused into the nodal-plane. In this case, uzj j ¼ 0 in the
nodal-plane (z = 0) and the acoustic radiation force
keeps the particles inside the nodal-plane without
modifying particle motion inside the nodal-plane in any
way. Therefore, when this situation is established, the
y-component of the Rayleigh streaming rules the dynamics
of the particles.
Actually, Rayleigh streaming also consists in a centrif-
ugal motion from directly under the transducer, as observed
on the Fig. 10a. The motion is actually applied only to the
solvent, not the particles. Through the Stokes force, the
particles follow the streaming since their inertia has been
chosen to be good passive tracers. The particles just leave
the region directly under the transducer and aggregate at
the periphery. This phenomenon happens only after a long
time, when most of the particles have already focused in
the nodal-plane, and only for the in-plane particles. The
expression for the Rayleigh velocity (Spengler et al. 2003)
is:
uR ¼3pv2
0 fac
64c20
l; with EAh i ¼ q0v20
2ð9Þ
where v0 is the acoustic velocity amplitude, fac the acoustic
frequency and l the channel width. From Eq. 9, one can see
that measuring uR gives access to one of the unknown
parameters, v0, and as a consequence, EAh i; which is very
useful for some numerical calculations about the acoustic
focusing. This is precisely the way EAh i has been estimated
in Sect. 3.1.
In the perspective of micro-PIV measurements, and then
in the short time limit, we will avoid the Rayleigh
Fig. 9 z-component uz (y, z) of the Rayleigh microstreaming between
parallel walls. This graphic corresponds to Eq. (8). y 2 � l2; l
2
and
z 2 �h2; h
2
Microfluid Nanofluid (2009) 7:857–867 865
123
streaming. This is the reason why very energetic acoustic
waves are not advised even if it maximizes the acoustic
focusing. In Fig. 10a, the order of magnitude of the
acoustic streaming is uR = 36 lm s-1. Therefore, for
P = 10 V, acoustic streaming would not affect the main
flow as long as Q� QR ¼ 0:14 ml h�1ðQR being the flow
rate corresponding to uR). The flow rates used in our
experiments are higher than QR, therefore Rayleigh
streaming does not affect the main flow. However, Ray-
leigh streaming could have an effect on the velocity field if
P was a lot higher and Q the same, or even lower.
After some times, the particles aggregate at the periph-
ery, increasing the local concentration, and some clumps
are rapidly formed. This phenomenon is the consequence
of particles interactions (Nyborg 1978). Indeed, now that
the particles are concentrated only in the nodal-plane, the
local particle concentration has strongly increased. There-
fore, Bjerknes forces (or secondary radiation forces
[Bjerknes 1906]) cause attraction between particles, which
leads to the creation of clusters of particles. Then, the
particles aggregates form spiral structures, as shown on
Fig. 10b. Of course, this process can only be observed
without mean flow and in the long time limit, after all the
particles have focused in the nodal-plane. In the perspec-
tive of PIV measurements we will always avoid this situ-
ation since an homogeneous distribution of particles is
important for the auto-correlation calculation. This aggre-
gation process could not be observed together with a
mean flow.
7 Conclusion
Acoustic focusing of particles has been validated and
quantified as a function of different parameters. The
method is based on a radiation acoustic force, generated
from a standing acoustic wave in the micro-channel, which
moves the particles toward the measurement plane and
therefore eliminates most of the out-of-plane particles that
generate background noise. It was shown that the dynamics
of the focusing depends strongly on the particles concen-
tration, the pressure amplitude and the particles radius.
Some counterparts are to be taken into account, including
some after-effects of the acoustic waves, especially
acoustic streaming and aggregation. It has been shown that
it can help improving the PIV measurements but it can also
be used, from a general point of view, to manipulate or sort
particles in a micro-channel or in Hele-Shaw cells (Hoyos
et al. 2005).
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